MA2033-Linear Algebra-L2S4-2013/14-www.math.mrt.ac.lk/UCJ-20140424
Definition: (,∗) is a Group ⇔
1. ≠ ∅ : set is non empty
2. ∀ , ∈ ; ∗ ∈ : binary operation is close
3. ∀ , , ∈ ; ∗ ( ∗ ) = ( ∗ ) ∗ : operation is associative
4. ∃ ∈ , ∀ ∈ ; ∗ = ∗ = : there is an identity element
5. ∀ ∈ , ∃ ∈ ; ∗ = ∗ = : each element has its inverse element
Example: Which of the following are groups?
1. (ℝ, +): set of real numbers with addition
2. (ℝ,⋅): set of real numbers with multiplication
3. ({1,0}, +) : Boolean algebra with OR
4. ({1,0},⋅) : Boolean algebra with AND
5. (ℝ× , +): set of 2 × 2 matrices with real entries with addition
6. (ℝ× ,⋅): set of 2 × 2 matrices with real entries with multiplication
7. (ℝ): set of invertible × matrices under multiplication
8. set of invertible functions under composition.
9. set of 4th roots of 1 under multiplication
10. !: symmetry group of equilateral triangle under composition
11. " = {(#, $): $ = # ! + &# + ', 4&! + 27' ≠ 0} ∪ {+}: elliptic curve group, +: point at infinity
Theorem: (,∗) is a group ⇒
1. is unique
2. is unique
3. ∀ ∈ ; . =
4. ∀ , ∈ ; ∗ = ∗
5.
∗ = ∗ ⟹ = : right cancellation
6.
∗ = ∗ ⟹ = : left cancellation
Definition: (,∗) is an Abelian Group ⇔
1. (,∗) is a group
2. ∀ , ∈ ; ∗ = ∗ : commutative
Example: which of the above examples are Abelian Groups?
Definition: (0, +,⋅) is a Field ⇔
1. (0, +) is an Abelian Group
2. (0\{0},⋅) is an Abelian Group: 0 is the identity for +
3. ∀ , ∈ 0; ⋅ ∈ 0: closed under ∙
4. ∀ , , ∈ 0; ⋅ ( + ) = ( ⋅ ) + ( ⋅ ): distributive
Note : we also use the following
1. − is the inverse of for +
2. 45 is the inverse of for ∙
3. 1 is the multiplicative identity for ∙
Example: See which of the following are fields
1. (ℝ, +,⋅)
2. (ℝ,∙, +)
3. (ℂ, +,⋅): set of complex numbers with addition and multiplication
4. (ℚ, +,⋅): set of rational numbers with addition and multiplication
5. ℚ8√2: = { + √2 ∶ , ∈ ℚ}
6. ({1,0}, +,⋅)
7. ({<, 0},∧,∨) : set of logic with AND and OR
8. ( (ℝ), +,∙)
9. (?,∪,∩): universal set with union and intersection
10. AB : finite field with prime C elements under DEF C addition and multiplication
Theorem: If C is prime and is an integer then B ≡
Theorem: (0, +,⋅) is a Field ⇒ ∀ ∈ 0; 0 ⋅ = 0
(DEF C)
MA2033-Linear Algebra-L2S4-2013/14-www.math.mrt.ac.lk/UCJ-20140424
Definition: (H,∗,∘) is a vector space over (0, +,⋅)
1. (H,∗) is an Abelian Group
2. (0, +,⋅) is a Field
3. ∀ ∈ 0, ∀# ∈ H; ∘ # ∈ H: closed under scalar multiplication
4. ∀ , ∈ 0, ∀# ∈ H; ( + ) ∘ # = ( ∘ #) ∗ ( ∘ #) : distributive w.r.t. scalar addition
5. ∀ ∈ 0, ∀#, $ ∈ H; ∘ (# ∗ $) = ( ∘ #) ∗ ( ∘ $) : distributive w.r.t. vector addition
6. ∀ , ∈ 0, ∀# ∈ H; ( ⋅ ) ∘ # = ∘ ( ∘ #) : associative w.r.t. scalar product
7. ∀# ∈ H; 1 ∘ # = #
Note: the above four operations are functions such that
+: 0 × 0 → 0: scalar addition
⋅: 0 × 0 → 0: scalar product
∗: H × H → H: vector addtiopn
∘: 0 × H → H: scalar multiplication
Example: which of the above examples are Vector Spaces?
1. (ℝ , +,⋅) over (ℝ, +,⋅): set of vectors with real entries
2. (ℝ, +,⋅) over (ℝ, +,⋅)
3. (ℂ, +,⋅) over (ℝ, +,⋅)
4. (ℝ, +,⋅) over (ℂ, +,⋅)
5. (ℂ, +,⋅) over (ℂ, +,⋅)
6. (ℝ[#], +,⋅) over (ℝ, +,⋅): set of polynomial in # with real coefficients
7. (H, +,⋅) over (ℝ, +,⋅): set of solution of a linear differential equation
8. (M[0,1], +,⋅) over (ℝ, +,⋅): continuous functions on [0,1]
Theorem:
(H,∗,∘) over (0, +,⋅) is a vector space ⇒
1. ∀# ∈ H; 0 ∘ # = 2. ∀ ∈ 0; ∘ = 3. ∀ ∈ H, ∀ ∈ 0; (− ) ∘ # = (
∘ #)
4.
∘ # = ⇒ = 0 EN # = Definition:
(O,∗,∘) is a sub vector space of (H,∗,∘) over (0, +,⋅) ⟺
1. (H,∗,∘) is a vector space over (0, +,⋅)
2. O ⊆ H
3. (O,∗,∘) is a vector space over (0, +,⋅)
Theorem:
1. (H,∗,∘) is a vector space over (0, +,⋅)
2. O ⊆ H: subset
3. ∀#, $ ∈ H; # ∗ $ ∈ H: closed under addition
4. ∀ ∈ 0, ∀#, $ ∈ H; ∘ # ∈ H: closed under scalar multiplication
⇒
1. (O,∗,∘) is a vector space over (0, +,⋅)
2. (O,∗,∘) is a sub vector space of (H,∗,∘) over (0, +,⋅)
Note: When there is no threat of confusion we may use the following
1. # ∗ $ = # + $
2.
⋅ =
3.
∘# = #
4. #̅ = −#
5. = 0
Example: Write the rules for vector space with this new notation
H is a vector space over 0 (or H is a 0-vector space)
' ⊆ H, ST ∈ ', T ∈ 0
Definition: ∑T
T ST
is a linear combination of '